arxiv:0809.3397v1 [astro-ph] 19 sep 2008surprisingly,all molecular gas tori have high velocity...

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Astronomy & Astrophysicsmanuscript no.(will be inserted by hand later)

Starbursts and torus evolution in AGN

B. Vollmer1, T. Beckert2, and R.I. Davies3

1 CDS, Observatoire astronomique de Strasbourg, UMR 7550, 11, rue de l’universite, 67000 Strasbourg, France2 Max Planck Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany,3 Max Planck Insitut fur extraterrestrische Physik, Postfach 1312, 85741, Garching, Germany

Received / Accepted

Abstract. Recent VLT SINFONI observations of the close environments (∼ 30 pc) of nearby AGNs have shown that thickgas tori and starbursts with ages between10 and150 Myr are frequently found. By applying these observations toa previouslyestablished analytical model of clumpy accretion disks, wesuggest an evolutionary sequence for starburst and AGN phases.Whereas the observed properties of the gas tell us about the current state of the torus, the starburst characteristics provideinformation on the history of the torus. In the suggested evolution, a torus passes through 3 different phases predetermined byan external mass accretion rate. Started by an initial, short, and massive gas infall, a turbulent and stellar wind-drivenQ ∼ 1disk is formed in which the starburst proceeds. Once the supernovae explode the intercloud medium is removed, leaving amassive, geometrically thick, collisional disk with a decreasing, but still high-mass accretion rate. When the mass accretion ratehas significantly decreased, the collisional torus becomesthin and transparent as the circumnuclear disk in the Galactic centerof the Milky Way. Variations on this scenario are possible either when there is a second short and massive gas infall, in whichcase the torus may switch back into the starburst mode, or when there is no initial short massive gas infall. All observed tori upto now have been collisional and thick. The observations show that this phase can last more than100 Myr. During this phasethe decrease in the mass accretion rate within the torus is slow (a factor of 4 within150 Myr). The collisional tori also formstars, but with an efficiency of about10% when compared to a turbulent disk.

Key words. Galaxies: active – Galaxies: nuclei – ISM: clouds – ISM: structure – ISM: kinematics and dynamics

1. Introduction

In the unification scheme for active galactic nuclei (AGN) thecentral massive black hole is surrounded by a geometrical thickgas and dust torus (see, e.g., Antonucci 1993). If the observer’sline-of-sight crosses the torus material, the AGN is entirely ob-scured from near-IR to soft X-rays and only visible at X-ray en-ergies if the gas column density is not too high (Seyfert 2 galax-ies). On the other hand, if the torus is oriented face-on withrespect to the observer, the central engine is visible (Seyfert1 galaxies). The spectral energy distributions (SEDs) of mostquasars and AGN in Seyfert galaxies have a pronounced sec-ondary peak in the mid-infrared (mid-IR) (e.g. Sanders et al.1989; Elvis et al. 1994), which is interpreted as thermal emis-sion by hot dust in the torus. The dust is heated by the primaryoptical/ultraviolet (UV) continuum radiation, and the torus ex-tends from the dust sublimation radius outwards (Barvainis1987).

The geometrical thickness of the torus in the gravitationalpotential of the galactic nucleus implies a vertical velocity dis-persion of about50-100 km s−1. If one assumes that the diskis continuous, i.e. thermally supported, this correspondsto atemperature of∼ 105 K. Since this is beyond the dust subli-

Send offprint requests to: B. Vollmer, e-mail: [email protected]

mation temperature (∼ 103 K), thick tori have to be clumpyor must be supported by additional forces other than thermalpressure. Krolik & Begelman (1988) proposed a clumpy torusmodel where the clumps have supersonic velocities. Vollmeretal. (2004) and Beckert & Duschl (2004) elaborated this modelin which orbital motion can be randomized if magnetic fieldspermit the cloud collisions to be sufficiently elastic. Vollmeret al. (2004) found that the circumnuclear disk (CND) in theGalactic center (Gusten et al. 1987) and obscuring tori sharethe same gas physics, where the mass of clouds is in the range20 - 50M⊙ and their density close to the limit of disruptionby tidal shear. A change in matter supply and the dissipationofkinetic energy can turn a torus into a CND-like structure andvice versa. Any massive torus will naturally lead to sufficientlyhigh mass accretion rates to feed a luminous AGN.

If and how efficient these clumpy tori form stars is an openquestion. The large majority of observational studies probedthe nuclear star formation on scales of a few hundred parsecs(see, e.g., Sarzi et al. 2007, Asari et al. 2007, Gonzales Delgado& Cid Fernandes 2005, Cid Fernandes et al. 2004). These stud-ies resulted in a general view that about30% - 50% of thesample AGNs are associated with recent (ages less than a few100 Myr) star formation on these scales. Thanks to the highspatial resolution of the near-infrared adaptive optics integralfield spectrograph SINFONI, it has only recently become pos-

http://arxiv.org/abs/0809.3397v1

2 Vollmer, Beckert, & Davies: Starbursts and torus evolution in AGN

sible to study the environments of AGN on the 10 pc scale.Davies et al. (2007) analyzed star formation in the nuclei ofnine Seyfert galaxies at spatial resolutions down to0.085′′.They found recent, but no longer active, starbursts in the centralregions which occurred10 - 300 Myr ago. Moreover, Hicks etal. (2008) were able to measure the rotation and dispersion ve-locity of the molecular gas in these galaxies using the2.12 µmH2 (1-0)S(1) line. Surprisingly, all molecular gas tori have highvelocity dispersions and are therefore geometrically thick.

In this article we compare the observations of Davies et al.(2007) and Hicks et al. (2008) with the expectations of ana-lytical models of clumpy accretion disks developed in Vollmer& Beckert (2002, 2003) and Vollmer et al. (2004). In a firststep, we test whether these models are able to describe obser-vations. In a second step, these models allow us to investigatethe scenario hypothesized by Davies et al. (2007) where spo-radic, short-lived starbursts are due to short massive accretionevents in the central region, followed by more quiescent phasesuntil there is another episode of accretion. This scenario is cor-roborated by a closer look at the nucleus of NGC 3227 whereDavies et al. (2006) found signs of a past starburst (∼ 40 Myrago) and a presently quiescent gas torus with a Toomre param-eterQ > 1.

2. The theory of clumpy gas disks

Within the framework of Vollmer & Beckert (2002, 2003) andVollmer et al. (2004) clumpy accretion disks are divided intotwo categories: (i) turbulent and (ii) collisional disks. In case(i) the ISM is regarded as a single entity which changes phase(molecular, atomic, ionized) according to internal (gas density,pressure, magnetic field) and external (gravitation, radiationfield, winds) conditions. Energy is injected into a turbulent cas-cade at the driving length scale (large scale) and dissipated atthe dissipation length scale (small scale). We identify thedis-sipation length scale with the characteristic size of selfgravitat-ing clouds. These clouds decouple from the the turbulent cas-cade and constitute the first energy sink. The source of energywhich is injected at the driving scale to maintain turbulence canbe either (i) mass accretion in the gravitational potentialof thegalactic center (fully gravitational FG model) or (ii) supernovaexplosions (SN model). In the collisional case energy is alsosupplied in the process of mass accretion in the gravitationalpotential of the galactic center and dissipated via partially in-elastic cloud–cloud collisions. The actual dissipation rate in in-dividual collisions is largely unknown. The disk evolutionismainly driven by the external mass accretion rate. Since thesemodels are equilibrium models, we assume that the mass ac-cretion rate is constant throughout the region of interest whenaveraged for a sufficiently long time (∼ Ω−1; hereΩ is theangular velocity of circular orbits in the gravitational potentialof the galatic nucleus) and maintained for at least the turnovertimescaleR/vturb of gas in the disk, whereR is the distancefrom the center of the galaxy andvturb the characteristic speedof turbulent eddies. All models give access to the global param-eters of the disk and the local parameters of the most massiveclouds (see Table 1). The free parameters of the models are theToomre parameterQ, the disk transparencyϑ (Eq. 7), and the

Table 1.Model parameters and their meaning

large scale disk

R galactic radiusvrot rotation velocityΩ angular velocityMdyn total enclosed (dynamical) massMgas total gas massvturb gas turbulent velocity dispersionQ Toomre parameterρ midplane densityΣ surface densityH disk height

M disk mass accretion rateν viscosityldriv turbulent driving length scaleδ scaling parameter between driving

scale length and cloud sizeΦV cloud volume filling factorζ viscosity scaling parametertHff disk vertical free fall timeρ∗ star formation rate per unit volume

Σ∗ star formation rate per unit surface

M∗ star formation rateξ conversion factor for SN energy fluxη star formation efficiency in the

collisional model

γ linking factor between Mgas and Mfor torus evolution

ϑ disk transparency

MBH mass accretion rate onto the black hole

Mwind wind outflow rateJUV AGN UV radiation field

small scale clouds

tcoll timescale for cloud− cloud collisionslcoll cloud mean free pathMcl cloud massrcl cloud radiusNcl cloud surface densitycs local sound speed within the cloudsts sound crossing time of cloudstclff cloud free fall timeT temperatureci sound crossing time of the ionized gas

mass accretion rateM . Other parameters are fixed using theGalactic values (Vollmer & Beckert 2002, 2003). Each modelhas an associated star formation rate. In the following we de-scribe these models in more detail.

2.1. Turbulent disks

In Vollmer & Beckert 2002 (Paper I) we developed an analyti-cal model for clumpy accretion disks and included a simplifieddescription of turbulence in the disk. In contrast to classical ac-cretion disk theory (see, e.g., Pringle 1981), we eliminated the“thermostat” mechanism, which implies a direct coupling be-tween the heat produced by viscous friction and the viscosity

Vollmer, Beckert, & Davies: Starbursts and torus evolutionin AGN 3

itself. The viscosity is usually assumed to be proportionaltothe thermal sound speed. Thus, the (gas) heating rate dependsitself on the gas temperature. This leads to an equilibrium cor-responding to a thermostat mechanism. Instead, we use energyflux conservation, where the potential energy that is gainedthrough mass accretion and differential rotation is cascaded byturbulence from large to small scales and dissipated there.

One fundamental approximation is that the kinetic energyis dissipated (removed from the turbulent cascade) when thegas clouds become self-gravitating. Turbulence transferstheenergy from the driving wavelengthldriv to the dissipationwavelengthldiss, which corresponds to the size of the largestselfgravitating clouds. The two length scales are linked bythescaling parameterζ. For a Kolmogorov-like turbulent energyspectrumζ = (ldriv/ldiss)

3

4 . In addition, the modeled diskshave a constant Toomre-Q parameter:

Q =vturb Ω

πGΣ≥ 1 , (1)

wherevturb is the turbulent velocity,Ω the angular velocity,Gthe gravitational constant, andΣ the gas surface density of thedisk. If we can approximate the total gas mass within a radiusR byMgas = πR2Σ the Toomre parameter can be rewritten

Q =vturbvrot,K

Mdyn

Mgas

, (2)

whereMdyn is the total enclosed mass andvrot the Keplerianrotation velocity.

Furthermore, we use the following prescription for the vis-cosity:

ν = ζ−1vturbldriv . (3)

We obtained a set of equations with 3 free parameters: theToomre parameterQ, the scaling parameterζ > 1, and themass accretion rate within the diskM . The mass accretion rateis the gas mass transported per time through the gas disk fromlarge to small radii. It must be supplied from the galaxy at theouter radius of the turbulent gas disk and is constant at all radiiin the disk. This set of equations can be solved analyticallyandthe results were already used in Paper I to describe properties ofour Galaxy. The solutions depend on the parametersQ, ζ, M ,vrot, andR. It turned out that the driving wavelength equals thedisk heightldriv = H .

In a second step we included the energy input due to su-pernova (SN) explosions (Vollmer & Beckert 2003, Paper II).The energy flux provided by SNe is transfered by turbulenceto smaller scales where it is again dissipated. The SN energyflux is assumed to be proportional to the local star formationrate. The local star formation rateρ∗ is taken to be proportionalto the mean density and inversely proportional to the local freefall time of the clouds. These clouds have sizes that are a factorδ smaller than the driving length scale. The factor of propor-tionality is the probability to find a self-gravitating cloud, i.e.the volume filling factor. The integration length in the verticalz direction is assumed to be the turbulent driving scale length,i.e. the vertical height in the disk where self-gravitatingcloudsare found:Σ∗ = ρ∗ ldriv. The SN energy per unit timeESN per

area∆A is therefore proportional to the local star formationrateΣ∗:

ESN

∆A= ξΣ∗ , (4)

where the factor of proportionalityξ is independent of the ra-dius in the disk. Its normalization with Galactic values yieldsξ = 4.6 10−8 (pc/yr)2.

In the FG model the energy transported through the turbu-lent cascade is supplied by mass accretion, which leads to anenergy flux equation of the form

ρνv2turbldriv

= −1

2πMvrot

∂Ω

∂R. (5)

In the case of SN driven turbulence the energy flux is deter-mined by the star formation rate

ρνv2turbldriv

= ξ Σ∗ . (6)

Furthermore, we take into account that the clouds are formedduring the interaction between SN remnants at the compressededges. The size of the cloudslcl is consequently smaller thanthe turbulent driving wavelength and we useδ = ldriv/lcl withδ ≥ 1.

2.2. Collisional disks and the stability of the clouds

If the selfgravitating clouds are stable, their collisionswill giverise to angular momentum redistribution again described byan effective viscosity. An equilibrium disk can be formed ifthere are fragmenting collisions or partially elastic collisions(the clouds are supposed to be magnetized). If the collisionaltimescaletcoll is longer or equal to the dynamical timescale,the resulting viscosity can be written as

ν = ϑ−1vturbH , (7)

where the disk transparencyϑ = tcollΩ > 1 andH is the diskheight. It follows that the collisional energy dissipationrate is

∆E

∆A∆t= f

Σv2turbtcoll

= fΣv3turblcoll

= fΣv3turbϑH

. (8)

Here the factorf , which has been omitted in Vollmer et al.(2004), accounts for the mean fraction of cloud mass partici-pating in the highly supersonic cloud collisions. For constantdensity clouds Krolik & Begelman (1988) argue thatf = 0.2.For more centrally condensed, self-gravitating clouds we willuse a factorf = 0.1 in this paper. This geometric factorfnot only reduces the average energy dissipation rate in cloudcollisions but also the angular momentum redistribution inthecollisions. The reduced energy dissipation is accompaniedbya correspondingly reduced mass accretion rateM for the sametransparency in the model disksν → fν in Eq.(7).

Since in the FG modelldriv = H , the FG model and thecollisional model are formally equivalent. We can thus useEq. (1 - 3) replacingζ by ϑ to describe the collisional model.Nonetheless the interpretation ofζ andϑ is completely differ-ent.


The cloud sizercl and the volume filling factorΦV ofclouds can be derived using their mean free path (see Vollmeret al. 2004)

lcoll = ϑH =4rcl3ΦV

(9)

and the fact that the clouds are selfgravitating

tclff =

√

3πΦV

32Gρ= ts =

rclcs

, (10)

wheretclff is the free fall timescale within clouds,ρ the diskoverall gas density,ts the sound crossing timescale, andcs thesound speed. This leads to

rcl =π2

8

Qc2sΩvturbϑ

(11)

and

ΦV =π2

6

Qc2sv2turbϑ

2. (12)

The cloud mass is then

Mcl =4π

3Φ−1

V ρr3cl =2

1

3 π4

64c4sϑ

−4

3G−4

3Q2

3 f1

3 M−1

3Ω−1 .(13)

These clouds move supersonically within the intercloudgas. If the intercloud gas is ionized, typical Mach numbers areabout 10. Therefore, we expect that the clouds might be de-stroyed by Rayleigh-Taylor instabilities. However, Rodiger &Hensler (2008) showed that these instabilities are suppressed inthe presence of (i) a sufficiently strong gravitational fieldof theclouds or (ii) a strong magnetic field. The ram pressure exertedon the clouds ispram = ρintv

2cl, whereρint is the intercloud

gas density andvcl is the cloud velocity. Following Rodiger &Hensler (2008) Rayleigh-Taylor instabilities are suppressed ifthe gravitational accelerationg is higher than the accelerationdue to the drag by ram pressureaD. This yields approximately:

g ∼MclG

r2cl> aD ∼

pramΣcl

. (14)

For the intercloud gas density we use the value of the GalacticCenterρint = 103 cm−3 (Erickson et al. 1994). UsingMcl =10 M⊙, rcl = 0.02 pc,vcl = 100 km s−1, the gravitational ac-celeration is about 4 times higher than the ram pressure drag.If the intercloud gas has a 10 times higher density, only a mag-netic field with a field strength ofB ∼

√

10 ρintv2cl ∼ 1 mGcan stabilize the clouds. This kind of field strength is observedin the Circumnuclear Disk in the Galactic Center (Plante et al.1995). It is thus plausible that the torus clouds are stable againstRayleigh-Taylor instabilities.

3. Star formation in clumpy gas disks

3.1. Turbulent disks

Following Paper II we assume that the star formation rate isproportional to the mean density of the disk and the inverse of

the characteristic timescale for the cloud collapse, i.e. the non-averaged local free fall timetclff :

ρ∗ ∝ρ

tclff. (15)

Sincetclff ∝ ρ−1

2 this corresponds to a Schmidt law of the formρ∗ ∝ ρ

3

2 . The factor of proportionality is given by the probabil-ity to find a self-gravitating cloud, i.e. the volume filling factorφV. Thus, the star formation rate is given by

ρ∗ = φV

ρ

tclff=

√

φV

ρ

tHff. (16)

Furthermore, we assume that stars are only born in the mid-plane of the disk in regions that have the size of the turbulentdriving length scaleldriv, because the clouds can collapse onlywithin the turbulent timescaletturb = ldriv/vturb. We thus ob-tain

Σ∗ = ρ∗ ldriv (17)

for the mass surface density turned into stars.

3.2. Collisional disks

For the collisional disk we assume that the star formation rateis proportional to the overall densityρ and the cloud collisionfrequencyt−1

coll = ϑ−1Ω:

ρ∗ = ηρϑ−1Ω , (18)

whereη is an a priori unknown efficiency factor. In terms ofstellar mass per time this gives

M∗ = ηMgasϑ−1Ω . (19)

4. Thick disks in a generic galactic center

The theoretical model described in the previous sectionscan now be applied to recent near-IR high-spatial resolutionSINFONI observations of nearby AGN. Davies et al. (2007)showed that there had been recent star formation in the cen-tral few tens of parsecs in a sample of nearby AGN. Followingon from this, Hicks et al. (2008) showed that distribution andkinematics of the central concentrations of gas were simi-lar to those of the stars, and was geometrically thick withvturb/vrot > 1/2. They argued that this gas comprised thelarge scale structure of the tori, implying that tori can formstars. And they gave an estimate for the gas mass in this regionas 10% of the dynamical mass. Before we derive the physicaldisk parameters for the individual AGNs observed by Davieset al. (2007), we give an overview over the different types ofthick tori. Note thatϑ is the disk transparency, i.e.ϑ ≤ 1 im-plies an opaque disk whereas a largeϑ results in a transpar-ent disk. For a generic galactic center we assume a dynamicalmass (which for radiiR larger than a few pc is dominated bythe stellar content), ofMdyn = 108 M⊙ and a gas mass ofMgas = 107 M⊙, both within a radius ofR = 20 pc. The en-closed mass leads to a rotation velocity of150 km s−1 at thisradius. In addition, we adopt a cloud internal sound speed of


cs = 1.5 km s−1. The value for the sound speed corresponds toa cloud temperature of∼ 500 K when only thermal gas pres-sure is considered. The sound speed is a measure of the pres-sure support against self-gravity and additional contributionsto the pressure gradient inside clouds like magnetic fields maycontribute. The adopted sound speed leads to cloud masses ofMcl ∝ c4s ∼ 10 M⊙. The disk is stable with a Toomre param-eter ofQ = 4.7 for a turbulent velocity supporting the verticalthickness ofvturb = 70 km/s. Moreover, we assume a star for-mation rate ofM = 0.1 M⊙yr−1. This can be considered anupper limit to the current star formation rate in the centralfewtens of parsecs. Davies et al. (2007) showed that while star for-mation had occured there recently, it has now ceased. Based onthe exponentially decaying starburst model they used and thetime averaged star formation rates they estimated, the currentrates are expected to be below this limit.

In the following we generate a generic set of models which(i) can describe different evolutionary phases of a∼ 20 pcscale gas disk in terms of mass accretion rate, disk thick-ness/transparency, and star formation rate and (ii) can repro-duce the observations at a generic time in the evolutionary se-quence. We first compare different disk models, representingevolutionary stages, to the observations of the current state ofthe disk (Sec. 4.1–4.5). It is shown that a massive accretionevent leads to a relatively thin turbulent disk which forms starsat a high rate (SN model). Once the SN explode the intercloudmedium is blown out, leaving only dense, compact clouds. Thisresults in a collisional disk that may assume one of severalstates (Sec. 4.2–4.5) depending on the external mass accre-tion rateM , the ToomreQ, and transparencyϑ parameters. InSec. 5 we then investigate the evolution of the collisional disk.

4.1. A massive turbulent disk creates a starburst

We first try applying the turbulent SN disk model. With theparameters described above it would have a mass accretion rate

M = 2.18 M−1∗ M2

gasv4turbR

−2ξ−1 (20)

in excess of103 M⊙ yr−1 and a ratio between the turbulentdriving and dissipation length scales of

δ = 1.17 M−3∗ G3M6

gasvturbR−6ξ−2 > 104 . (21)

Such a model can be discarded, because its mass accretion rateis far too high with respect to any supply from the outer galaxyand the lifetimeMgas/M = 104 yr is shorter than the dynam-ical timeΩ−1 ∼ 105 yr.

On the other hand, if we assume that a turbulent SN disk isresponsible for the starburst with a star formation rate ofM∗ =1 M⊙yr−1 (comparable to the initial rate inferred by Davieset al. 2007) and that the turbulent velocity was lower (vturb =20 km s−1) than the present collisional disk, we find a massaccretion rate ofM = 2 M⊙yr−1 andδ = 23. These values areclose to the values for the large-scale Galactic disk (see PaperII). A typical cloud has a radius ofrcl = 0.1 pc and a mass ofMcl = 104 M⊙. This kind of disk contains about a thousandclouds withinR = 20 pc.

We conclude that a viable turbulent massive gas disk has arather low turbulent velocity and is therefore moderately thin

with H/R = vturb/vrot ∼ 0.13. It then yields large star for-mation rates of the order of one solar mass per year. This repre-sents a starburst which subsequently will destroy the disk oncethe supernovae explode after about 10 Myr. These explosionsdo not cause any harm to the densest and most massive clouds,but they clear the space between the clouds, i.e. they removetheinitial intercloud medium. We are then left with a collisionaldisk.

4.2. A massive, opaque, collisional disk(Q = 5, ϑ = 1)

The collisional disks can be distinguish by theirQ andϑ pa-rameters. We start with collisional and opaque disks, i.e. themean free path of the clouds is about the height of the diskH .This implies that along a vertical path through the disk thereis on average one intervening cloud. Along a path in the diskmidplane towards the center there are typicallyN ∼ 10 cloudsblocking the direct view. Collisions are frequent in such a torusor disk. This yields a mass accretion rate of

M = 2v3turbGQϑ

= 3.4 M⊙ yr−1 . (22)

The volume filling factor of the clouds isΦV = 0.004, theclouds have typical radii ofrcl = 0.02 pc, and masses ofMcl =15 M⊙.

4.3. A massive, transparent, collisional disk(Q = 5, ϑ = 10)

The large mass accretion rate in the above model (Sec. 4.2) de-pends linearly on the collision rate and is a consequence of thelow transparency. If the disk is more transparent,ϑ = 10, themass accretion rate is thereforeM = 0.3 M⊙yr−1, the volumefilling factor of the clouds isΦV = 4 10−5, and the clouds havetypical radii ofrcl = 0.002 pc and masses ofMcl = 1.5 M⊙.We see that for the same Toomre-Q a larger transparency im-plies smaller and less massive clouds with smaller volume fill-ing factors.

4.4. A light, opaque, collisional disk(Q = 50, ϑ = 1)

If clouds are large and less dense we can have a disk which islight but still optically opaque due to dust in the clouds. For thisdisk class we assume a gas mass of only1% of the dynamicalmass, i.e.Mgas = 106 M⊙. This yields a mass accretion rateof M = 0.3 M⊙yr−1. The volume filling factor isΦV = 0.04,and the typical cloud radii and masses arercl = 0.2 pc andMcl = 150 M⊙.

4.5. A light, transparent, collisional disk(Q = 50, ϑ = 10)

The last disk class is transparent,ϑ = 10, and has a small gasmass,Mgas = 106 M⊙. It has the lowest mass accretion rate,M = 0.03 M⊙yr−1, and a low volume filling factor,ΦV =


4 10−4. The cloud radii and masses are the same as those of themassive, opaque disk (rcl = 0.02 pc andMcl = 15 M⊙).

We conclude that (i) a high mass accretion leads to a mas-sive, opaque disk and (ii) a (Q = 5,ϑ = 1)-disk and a (Q = 50,ϑ = 10)-disk share clouds of the same mass and size. Theseclouds are very similar to those found in the Galactic Center(see Vollmer et al. 2004). We thus can draw up a disk evolutionin which a massive, opaque disk evolves with time into a light,transparent disk.

For the rest of the article we assume that the cloud massof all disks isMcl = 10 M⊙. This is equivalent to a commoncolumn density of all clouds of

Ncl =3

4

Ωvturbϑ

πGQ. (23)

A thick disk withϑ/Q = 1/5 has clouds with column densitiesof Ncl ∼ 1024 cm−2.

5. Torus evolution

The collisional disks described above can be identified withtheobserved30 pc-scale gas concentrations observed by Davies etal. (2007) and Hicks et al. (2008) which arguably correspondto the large scale structure of AGN tori.

It is now investigated how a collisional torus can evolvefrom a massive to a less massive state. To do so we assume thatthe gas mass of the torus is proportional to its mass accretionrate

Mgas = 2−1

3G−2

3ϑ1

3Q−2

3 f−1

3 M1

3 vrotR = γMx . (24)

In this way we subsume all possible time dependencies of var-ious parameters and their correlations in the time evolution ofM . Together with Eq. 13 this yields

ϑ =π4

64

c4sR2

γMclG2

1

Mx. (25)

The expressions for the Toomre parameter, the turbulent veloc-ity, and the star formation rate are then

Q =π2

√128

c2sR4Ω

3

2

γ2G2M1

2

cl f1

2

1

M2x− 1

2

, (26)

vturb =π2

√128

c2sR2Ω

1

2

γM1

2

clGf1

2

1

Mx− 1

2

, (27)

and

M∗ = η64

π4

Mclγ2G2Ω

c4sR2

M2x . (28)

In this way the behavior of the transparency, Toomre parameter,thickness of the disk viaH = vturb/Ω, and star formation ratecan be identified with the change of the external mass accretionrate. For our stationary equilibrium disks to be applicablewemust require the changes of the external mass accretion to beslow, so that the whole disk can adjust to the changing externalconditions. This time for adjustmentteq is approximately theratio between the torus gas mass and mass accretion rate. For

typical values ofMgas ∼ 107 M⊙ (Tab. 2) andM ∼ 1 M⊙yr−1

(Tab. 4) this leads toteq ∼ 10 Myr. This is smaller than the ob-served starburst ages (Tab. 2) for all AGNs except NGC 1097where the time of adjustment and the starburst age are compa-rable.

5.1. Evolution at constant thickness

Whenever the disk thickness stays the same during its evolu-tion, vturb =const. in time, and subsequentlyx = 1

2(Eq. 27),

this implies that the external mass accretion rate stays at ahighlevel. The gas mass, star formation rate,ϑ, and the ToomreQparameter depend on the mass accretion rateM in the follow-ing way:

Mgas ∝ M1

2 , M∗ ∝ M , ϑ ∝ M−1

2 , Q ∝ M−1

2 . (29)

The dependence of these parameters on the gas mass is then:

M∗ ∝ M2gas , ϑ ∝ M−1

gas , Q ∝ M−1gas (30)

and the relation for the cloud mass and the volume filling factorare:

Mcl = const. , ΦV ∝ Mgas . (31)

It is remarkable that the star formation rate is linearly coupledto the mass accretion rate in this scenario.

5.2. Evolution at constant mass

Alternatively, at later stages when the mass accretion rateis low and changes little, the torus mass may stay constant,Mgas =const, and subsequentlyx = 0 (Eq. 27). The turbu-lent velocity dispersion, star formation rate,ϑ, and the Toomredepend on the mass accretion rateM now in the following way:

vturb ∝ M1

2 , M∗ = const. , ϑ = const. , Q ∝ M1

2 . (32)

The relation for the cloud mass and the volume filling factorare:

Mcl = const. , ΦV ∝ M−1

2 . (33)

At constant disk mass and decreasing external mass supplythe disk will become geometrically thin without changing thetransparency.

An interesting result is that during both types of torusevolution—at constant turbulent velocity and at constant gasmass—the cloud mass does not change.

6. Torus evolution scenarios

In the picture of quasi-stationary equilibrium disks driven bycloud collisions the evolution will be determined by the ex-ternal mass accretion rate, i.e. the mass inflow from distances> 100 pc. We divide the torus evolution into three phases:

– Phase I: Initial massive infall and formation of a turbulent,massive gas disk:An initial rapid infall of a large amount of gas ,Mgas ∼


107 M⊙, within a short time (∆t < 1 Myr; M >10 M⊙yr−1) leads to the formation of a massive (Q ∼ 1),moderately thin (vturb/vrot < 5) gas disk in which star for-mation proceeds. This phase will be recognized as a star-burst. The disk becomes turbulent and the turbulence ismaintained by the energy input from stellar winds. After∼ 10 Myr the first SN explode and will rapidly removethe initial intercloud medium from the disk. Only the mostmassive and densest clouds which are not Jeans unstablewill survive. This leads in the following to acollisionaltorus.

– Phase II: Torus evolution at constant turbulent velocity:During the first phase of its evolution the massive colli-sional torus stays thick. This implies that the mass accretionrate within the torus, and thus also the external mass accre-tion rateM , do not decrease significantly during this phase.The gas mass and the cloud collision rate (tcoll ∝ ϑ−1) de-crease with decreasingM , whereasQ increases with thesquare root of the external mass accretion rate. The star for-mation rate within the torus decreases with decreasingM(Eq. 30).

– Phase III: Torus evolution at constant gas mass:Once the external mass accretion rate has significantly de-creased, the torus evolves at constant gas mass. The veloc-ity dispersion andQ decrease with the square root of theexternal mass accretion rate, whereas the cloud collisionrate (tcoll ∝ ϑ) and the star formation rate stays constant(Eq. 32). The Circumnuclear Disk (CND) in the GalacticCenter (Gusten et al. 1987) represents such a late stage(Q = 190, ϑ = 15; Vollmer et al. 2004) of the torus todisk evolution.

Depending on the time evolution of the external mass accre-tion rate (from distances larger than100 pc) we suggest threepossible evolution scenarios (Fig. 1):

– Scenario I:The torus never reaches Phase I, because (i) it is alreadyclumpy from the very beginning, (ii) there is already starformation occurring at scales of∼ 10 pc from the galacticcenter; the associated SN explosions and/or winds removethe existing intercloud medium and/or inhibit the forma-tion of an intercloud medium. The two possibilities implythat the initial mass accretion rate is not very high. A thirdpossibility (iii) is that the disk’s velocity dispersion ispro-hibitively large to allow aQ = 1 disk. In this case star for-mation proceeds via cloud-cloud collisions. The star forma-tion rate is lower than that of a massive turbulent disk. Therate of momentum injection due to SNe and stellar winds is

F = p∗∆A = 5 1033(M∗

M⊙yr−1) dyne , (34)

wherep∗ is the pressure due to SNe and stellar winds ex-erted on a surface∆A (Veilleux et al. 2005). Assuming thesame initial turbulent velocity for the clouds and the in-tercloud medium, the intercloud medium can be removedfrom the disk ifp∗ ≥ ρIMΦIM

V v2turb, whereρIM andΦIMV are

the density and the volume filling factor of the intercloud

medium. AssumingM∗ = 0.02 M⊙yr−1, R = 10 pc(Sect. 7.5), andvturb = 50 km s−1 (Tab. 2) leads tonIMΦIM

V ≤ 103 cm−3. Since the disk volume averageddensity of the clouds isnclΦV = 104 cm−3 (Tab. 4), theintercloud space can only be cleared by SN explosions andstellar winds if the intercloud medium contains less than∼ 10% of the total disk mass.

– Scenario II:Due to an initial, massive infall a massive (Q ∼ 1) tur-bulent star-forming disk is formed (Phase I). The turbu-lence in this disk is maintained through the energy sup-ply by feedback from rapid star formation. The subsequentSN explosions destroy the disk structure after10 Myr, i.e.the intercloud medium is removed leaving only the densest,most massive clouds which remain Jeans-stable. The diskbecomes collisional and stays geometrically thick (PhaseII). After ∼ 100 Myr the mass accretion rate decreases andthe disk becomes thin (Phase III) and ultimately transparent(ϑ > 5). The time at which the torus changes from Phase IIinto Phase III depends on the time evolution of the externalmass accretion rate in this scenario.

– Scenario III:Due to an initial massive infall event, a massive (Q ∼1) turbulent star-forming disk appears (Phase I). As inScenario II, SN explosions destroy the disk structure after∼ 10 Myr, the intercloud medium is removed and only thedensest, most massive clouds are left over, which are Jeans-stable. The disk becomes collisional and will stay thick aslong as the external mass accretion rate is sufficiently high(Phase II). Due to a secondary massive and rapid gas infalla second massive (Q ∼ 1) turbulent star-forming disk canform (Phase I), which evolves again into a collisional diskafter∼ 10 Myr (Phase II). After∼ 100 Myr the mass ac-cretion rate has sufficiently decreased and the disk becomesthin (Phase III) and ultimately transparent (ϑ > 5). Thetime at which the torus changes from Phase II into PhaseIII depends again on the time evolution of the external massaccretion rate.

7. Applying the observations

The scenarios of Sect. 6 derived from our analytical modelscan now be compared with the VLT SINFONI observations ofDavies et al. (2007) and Hicks et al. (2008). As has been shownin Sect. 4.1 the present state of the disk can only be describedconsistently by a collisional disk model. The comparison withobservations will allow us to derive the parameters of (i) thepresent torus, (ii) the initial torus immediately after Phase I,(iii) the massive (Q ∼ 1) turbulent gas disk that gave rise to theinitial starburst (scenario II and III), and (iii) the star formationefficiency of the collisional phase.

The observables, i.e. the input parameters for our analyticalmodel, are the radius from the galactic centerR, the rotationvelocity vrot, the turbulent velocity dispersionvturb, the gasmassMgas, the peak star formation rate during the initial star-burstMpeak

∗ , and the age of the initial starbursttSB. We assumefor all AGN, except NGC 1097, a starburst duration of10 Myr,


Fig. 1. Schematic of torus evolution scenarios: Torus mass ac-cretion rate (M⊙yr−1) is plotted as a function of time (Myr).Phase I: A massive (Q ∼ 1) turbulent disk is associated witha starburst. Phase II: A collisional torus evolves at a constantturbulent velocity dispersion, i.e. thick torus evolution. PhaseIII: The torus becomes thin and evolves at an approximatelyconstant gas mass.

i.e. the the starburst continues until the first SN explode. Weonly apply our model to a subsample of 6 nearby AGNs fromDavies et al. (2007) for which these input parameters are suf-ficiently well known.The parameters for these objects can befound in Table 2. For each object, the table gives its Seyferttype and the radius within which the subsequent parametersapply. The dispersionvturb is the value measured from the data(after accounting for instrumental broadening), while therota-tion velocityvrot is a Keplerian equivalent value. This meansthat it represents the rotation velocity that would be needed ifordered circular motions in a single plane supported the entiredynamical mass. It is therefore significantly greater than themeasured rotation speed. The gas massMgas is difficult to de-rive. Hicks et al. (2008) estimated it from a combination of di-agnostics, including the 1.3 mm CO luminosity, the 2.12µm H2

1-0 S(1) luminosity, and a comparison to the gas-to-dynamicalmass ratios in other spiral and starburst galaxies. The peakstarformation ratesMpeak

∗ are derived from the starburst modelsused in Davies et al. (2007). They are simply the star formationrates required to form the young stars in a timescale of 10 Myr(in contrast to the time-averaged rates given in that paper). The

Table 3.Equations to derive the disk/torus properties from ob-servations. Velocities are in units of km s−1, radii in pc, massesin M⊙, and star formation rates in M⊙yr−1.

present collisional torus

Qpr = (vprturb

/vrot) (Mdyn/Mprgas)

ϑpr = 1.9× 10−9QprR/(vrotvpr

turbMcl)

Mpr = 2 (vprturb)3/(GQprϑpr)

massive turbulent gas disk

Qdisk = 9674 M−

2

11

∗ vrot

Mdiskgas = 3.97 × 105M

5

11∗ R

vdiskturb = 18.2 M3

11∗

initial collisional torus

M init = M(Mdiskgas /Mpr

gas)2

Qinit = QprMprgas/M

diskgas

ϑinit = ϑpr Mprgas/M

diskgas

only exception is NGC 3783, for which we have adopted a 2-starburst model with ages of 110 Myr and 30 Myr. Such modelswere not considered by Davies et al. (2007) because of the lim-ited number of diagnostics. In our scenario with stronger theo-retical constraints on the models we need these additional de-grees of freedom to reproduce the observations. The final col-umn gives the agetSB of the most recent starburst. All equa-tions to derive the properties of the present collisional torus, themassive(Q ∼ 1) turbulent gas disk, and the initial collisionaltorus are given in Table 3.

7.1. The present collisional torus

We recall the assumption that all gas clouds have a constantmass ofMcl = 10 M⊙. Moreover the sound speed within theclouds is set tocs = 1.5 km s−1. The ToomreQ parameterof the present disks is directly calculated from Eq. 1. We thenuse the following expressions forϑ and the mass accretion rateM which follow from the expression derived for the turbulentvelocity dispersion in Paper Ivturb = (1

2GϑQf−1M)

1

3 andEq. 13

ϑ =π4

64c4sG

−1QΩ−1M−1cl v−1

turb , (35)

and

M = 2fv3turbG−1Q−1ϑ−1 . (36)


Table 2. Input parameters from Davies et al. (2007) and Hicks et al. (2008); see Sect. 7 for details

name type R vrot vturb Mgas Mpeak∗ tSB

(pc) (km s−1) (km s−1) (106 M⊙) (M⊙yr−1) (Myr)

Circinus Sy2 10 93 56 2 0.02 80

NGC 3783 Sy1 30 88 35 5 0.3 30

NGC 3227 Sy1 30 173 95 20 5.0 40

NGC 1068 Sy2 30 184 88 20 14.0 200

NGC 1097 Sy1 30 122 59 10 1.0 8

NGC 7469 Sy1 30 123 63 10 3.0 110

Table 4.Parameters of the present collisional tori.

name M Q ϑ ΦV ρcl rcl(M⊙yr

−1) 10−3 107 (cm−3) (pc)

Circinus 0.75 5.7 1.9 1.2 3.9 0.01

NGC 3783 0.07 4.1 6.8 0.2 2.7 0.01

NGC 3227 4.27 5.5 1.7 0.5 3.9 0.01

NGC 1068 3.45 5.4 1.7 1.2 1.8 0.02

NGC 1097 0.59 4.8 3.4 0.4 2.6 0.02

NGC 7469 0.66 5.1 3.4 0.6 1.8 0.02

The torus mass accretion rates (Table 4) include the geomet-ric f = 0.1 factor introduced in Sec. 2.2. The resulting ratesare still higher than the black hole mass accretion rates andthe wind mass loss rates discussed in Sec. 8). The parametersderived in this way for our sample of 6 AGNs are shown inTable 4.

All tori show ToomreQ parameters between 4 and 6,i.e. they are massive thick tori. Moreover, 3 AGNs (Circinus,NGC 3227, and NGC 1068) have opaque tori (ϑ < 2), 2 AGNs(NGC 1097 and NGC 7469) show moderately transparent tori(ϑ ∼ 3 − 4), and 1 AGN (NGC 3783) has a transparent torus(ϑ > 5). It is worth noting that two out of the 3 AGNs withopaque tori are classified as Sy2. We find the smallest mass ac-cretion rate for NGC 3783. The highest mass accretion rates(NGC 3227 and NGC 1068) are several ten times higher thanthat of NGC 3783.

7.2. The massive (Q ∼ 1) turbulent gas disk

We now derive the parameters of the initial massive turbulentgas disk which gave rise to the initial starburst. For this weap-ply the SN model (see Sect. 2) where the energy source formaintaining turbulence in the disk are stellar winds. Theiren-ergy input is comparable to that of SN explosions (MacLow &Klessen 2004). Therefore we do not need to change the formal-ism of the SN model. We further assume that the mass accre-tion rate equals the peak star formation rate given by Davieset al. (2007) (Table 2) andδ = 5, which is the Galactic value(Vollmer & Beckert 2003).

We assume that for all AGNs, except for Circinus andNGC 3783, scenario II is valid. For Circinus we argue belowthat scenario I applies, because its peak star formation rate is

a factor of more than 10 lower when compared to the otherstarbursts. This does not imply that star formation is not oc-curring, just that there was no initial massive accretion event.For NGC 3783 scenario III is more applicable, because a singlestarburst leads to an enormously high initial mass accretion ratecompared to the present value, which we think is implausible.The double starburst we have adopted for NGC 3783, whichis consistent with the observations of Davies et al. (2007),re-quires a first intense burst withM∗ = 4.7 M⊙yr−1 to haveoccurred 110 Myr ago, followed by a second distinct burst withM∗ = 0.3 M⊙yr−1 only 30 Myr ago.

The ToomreQ parameter, the total gas mass, and the tur-bulent velocity dispersion can then be calculated using thefol-lowing expressions:

Q = 0.81G−2

11 M−

2

11

∗ δ−3

11 ξ−5

22 vrot , (37)

Mgas = 8.5 10−5G−8

11 M5

11

∗ δ−1

11 ξ1

11R , (38)

and

vturb = 0.82G3

11 M3

11

∗ δ−1

11 ξ1

11 , (39)

whereξ = 4.6 10−8 (pc/yr)2 (Vollmer & Beckert 2003).Using a peak star formation rate ofM∗ = 0.02 M⊙yr−1

for Circinus leads to a gas mass of the massive turbulent diskwhich is smaller than that of the present collisional torus.Wetherefore conclude that scenario II (Fig. 1) does not apply tothis AGN. Instead, scenario I yields more appropriate results.The nuclear disk in Circinus did not experience a turbulent,supernovae and stellar wind drivenQ ∼ 1 disk.

The gas masses of the turbulent starburst disks is between10% and100% higher than the gas mass of the correspond-ing present collisional torus, except for Circinus. In the courseof torus evolution (Fig. 1) the loss of gas mass in the disk ismoderate (up to a factor of 2; Tab. 2 and 5). The star formingdisks are moderately thin (vrot/vturb ∼ R/H ∼ 5-7) and theirToomreQ parameter is close to unity.

7.3. The initial collisional torus

Since all observed tori are thick (Table 2), they are all in PhaseII of their evolution (see Sect. 6), i.e. they evolve at constantthickness or velocity dispersion. This implies the following re-lation between the gas mass and the mass accretion rate:

M ∝ M2gas . (40)


Table 5. Parameters of the initial massive (Q ∼ 1) turbulentgas disk that gave rise to the initial starburst.

name Mgas M vturb Q(106 M⊙) (M⊙yr

−1) (km s−1)

Circinus 0.7 0.02 6 1.9

NGC 3783 6.9 0.3 13 1.1

NGC 3227 24.8 5.0 28 1.3

NGC 1068 40.0 14.0 37 1.2

NGC 1097 11.9 1.0 18 1.2

NGC 7469 19.6 3.0 25 1.0

Table 6.Parameters of the initial collisional tori.

name M Q ϑ(M⊙yr

−1)

Circinus 2.1 3.4 1.1

NGC 3783 0.1 3.0 5.0

NGC 3227 6.5 4.4 1.4

NGC 1068 13.5 2.7 0.9

NGC 1097 0.8 4.0 2.9

NGC 7469 2.6 2.6 1.7

We can safely assume that the gas mass of the initial collisionaltorus at the end of Phase I is close to the gas mass of the massive(Q ∼ 1) turbulent disk (Table 5). For Circinus we estimate thegas mass of the initial collisional torus by postulating that thedisk was initially opaque, i.e.ϑ ∼ 1. This leads to an initial gasmass ofMgas = 3.4 106 M⊙ at the beginning of Phase II.

For all other AGN a continuous transition from phase I to IIprovides the mass accretion rate of the initial collisionaltorus.TheQ andϑ parameters can then be calculated using the fol-lowing expressions (see Sect. 5):

Q ∝ M−1gas , ϑ ∝ M−1

gas . (41)

Four of the initial collisional disks at the beginning ofPhase II (Circinus, NGC 3227, NGC 1068, and NGC 7469)had ToomreQ parameters aroundQ = 3 andϑ close to unity,i.e. they were massive and opaque. Two initial collisional disks(NGC 3783 and NGC 1097) had been massive and transparent.

7.4. The evolution of the torus mass accretion rate

In this section we investigate the evolution of the torus massaccretion rate with time in Phase II (see Sect. 6). For this weplot in Fig. 2 the fraction between the mass accretion rate ofthe present collisional torus and that of the initial collisionaltorus (end of Phase I and beginning of Phase II) as a functionof time for our sample. To determine the time that a given toruspassed in Phase II, we place ourselves in scenario II (Fig. 1)andadopt the starburst ages of Davies et al. (2007) for all AGNsexcept NGC 3783. We assume scenario III for this galaxy, i.e.the occurrence of two distinct starbursts. This leads to an es-timated age of the most recent starburst of 30 Myr. The solid

Fig. 2.Ratio between the mass accretion rate of the present col-lisional torus and that of the initial collisional torus (end ofPhase I and beginning of Phase II) as a function of time. Thezero point in time is the end of Phase I (ϑ = 1 for Circinus).

line is meant to guide the eye. Based on this plot we concludethat the mass accretion rate shows a slow monotonic decreasewith time. It decreases to one fourth of its initial value in about150 Myr.

7.5. The star formation efficiency of collisional tori

Star formation in collisional disks is expected (see Sec. 3)to beproportional to the cloud collision ratet−1

coll = Ω/ϑ:

M∗ = ηMgasϑ−1Ω , (42)

whereη is the star formation efficiency which we would liketo determine. This is only possible for Circinus, because onlyin this case the peak star formation rate derived from obser-vations reflects the initial star formation rate of the collisionaltorus (scenario II). For all other AGNs the peak star formationrate is related to a massive (Q ∼ 1) turbulent gas disk. Duringthe torus evolution at constant thickness (Phase II) the star for-mation rate is proportional to the square of the gas mass

M∗ ∝ M2gas . (43)

The initial and present gas mass of the torus in Circinus areM init

gas = 3.4 106 M⊙ andMgas = 2 106 M⊙; the peak star for-

mation rate isM init∗ = 0.02 M⊙yr−1. This leads to an estimate

of the present star formation rate ofM∗ = 7 10−3 M⊙yr−1.The resulting star formation efficiency is

η =RM init

∗

M initgas vrot

≃ 10−3 . (44)

For a turbulent galactic disk the star formation law is

M∗ = ηgalMgasΩ . (45)

Thus, one has to compareη/ϑ ∼ 5 10−4 for a collisional diskwith ηgal = 0.017 derived for galactic SN driven turbulent gasdisk. We conclude that the star formation in a collisional torus


is about 10 times less efficient than that of a turbulent gas diskof the same mass. In our sample of 6 AGN the star formationrate of the present collisional torus is small compared to themass accretion rate. The average fraction between the mass ac-cretion rate and the star formation rate in these tori is25± 7.

7.6. Obscuring the starburst

The stellar luminosities calculated by Davies et al. (2007;theirFig. 8) indicate that on scales of several tens of parsecs, thestarbursts can only be weakly obscured. The reason for thisis explained by Davies et al. (2006), and summarised here.If one assumes no obscuration, the starbust already comprisestypically a few percent of the galaxy’s bolometric luminosity.Even a moderate optical depth will lead to nearly all the opti-cal light being absorbed and re-radiated in the far-infrared; andin addition the scaling one derives for the starburst would in-crease. The limiting constraint is that the starburst luminositycannot exceed the galaxy’s bolometric luminosity. We estimatethe resulting extinction (for screen models) to be in the rangeAV = 4 to 8. This is consistent with a clumpy torus with areafilling factors below unity (transparencyϑ > 1) for radii largerthan about1 pc. The optical extinction is then mainly due todust located in the foreground of the star forming region. InSy2 galaxies, however, not only the central AGN is obscuredby the torus, but also the inner part of the torus and embeddedstar forming regions are self-obscured by the extended torus. Toverify this idea, we use the model of Beckert & Duschl (2004)and Honig et al. (2006) which is based on the formalism devel-oped by Vollmer et al. (2004) to estimate the extinction using(i) a screen model where the starburst occurred in the torus mid-plane behind half the torus, and (ii) a mixed model where theforming stars and the clouds share the same spatial distributionwithin the torus. As an example we use parameters most appro-priate for NGC 3227 or NGC 1068:vturb/vrot = 0.55, ϑ = 2with an outer radius of 80 pc. Fig. 3 shows the mean number ofcloudsΛ along the line of sight through the torus for two torusinclinations in the case of a screen model.

Λ =

∫

l−1collds , (46)

wherelcoll = ϑH is the mean free path of the clouds andH thedisk height. Obscuration withΛ ≥ 3 leading to an likelihoodof non-obscuration of less thane−3 occurs at a radiiR ≤ 20 pcfor i = 60 and not at all fori = 40. For lower inclinationsthe obscuration by clouds is ineffective. Fori > 60 the ob-scuration pattern does not change much for geometrically thicktori.

Fig. 4 shows the ratio between the extinction-free starburstemission and the emission with obscuration by interveningclouds in the mixed case

I0Iext

=Λ

1− e−Λ. (47)

Complete absorptionI0/Iext ≫ 1 is only reached in the inner-most part of the AGN torus (R ≪ 1 pc).

The screen and the mixed models can be regarded as twoextreme cases and the reality maybe somewhere in between

-40 -20 0 20 40X [pc]

-40

-20

0

20

40

Z [p

c]

-40 -20 0 20 40X [pc]

-40

-20

0

20

40

Z [p

c]

Fig. 3. Projected mean number of cloudsΛ along the line ofsight through the torus for the screen model. Contour levelsare (0.5,0.75,1,1.5,2,2.5,3,..). The thick contour is forΛ = 1corresponding to an non obscuration probability ofe−1. Upperpanel: torus inclinationi = 45. Lower panel: torus inclinationi = 60 (i measured from the torus axis). The size of the boxis 100 pc.

these two models. Most probably the starburst will be signifi-cantly obscured at galactocentric radii smaller than1 pc. Theextinction of star formation at the 10 pc scale by torus cloudsis so low that additional extinction by extended dust lanes atlarger radii (> 100 pc) in the galaxies is possible. We can there-fore conclude that our torus model is consistent with the smallobserved extinction of the central starburst.


-40 -20 0 20 40X [pc]

-40

-20

0

20

40

Z [p

c]

Fig. 4. Ratio between the extinction-free starburst emission tothe emission with cloud absorptionI0/Iext for an torus incli-nation of i = 50. Contour levels are (1.5,2,2.5). The thickcontour is forI0/Iext = 2. It is the seconds outermost contour.

8. Fueling the central engine

The inner edge of the torus is thought to be set by the dustsublimation radius which is located at about0.1-0.5 pc fromthe central black hole. In Sect. 7 we have derived the mass ac-cretion rate of the tori, i.e. the mass transport rate arriving atthese inner edges. It is not clear what happens between the in-ner edge of the torus and the thin accretion disk around theblack hole and what the relation is to the maser emission re-gions. These maser disks have sizes of about0.1 pc. Between0.1 pc and0.5 pc from the central black hole an X-ray heatedwind is most likely formed (Krolik & Kriss 1995, 2001). Lessdense and sheared clouds lose their dust by evaporation at thesublimation temperature are consequently ionized by the AGNX-ray emission, heated, and blown away in a line-driven wind.What remains of the dust or is newly formed in the wind toobscure the AGN at even higher inclination angles is undeter-mined in this scenario. The mass accretion rate onto the centralblack holeMBH is thus the difference between the mass accre-tion within the torusM and the mass loss due to the AGN windMwind:

MBH = M − Mwind . (48)

Typical wind mass loss rates areMwind = 0.03-0.3 M⊙yr−1

(Blustin et al. 2005, 2007).To investigate the relation between the mass accretion rate

onto the central black hole and the present torus mass accretionrate, we plot their ratio as a function of the area filling factorΦA = 4/3ϑ−1. The mass accretion rate onto the central blackholeMBH is derived from the AGN luminosity

L = ηMBHc2 , (49)

Fig. 5. Ratio between the mass accretion rate onto the centralblack hole and the present torus mass accretion rate as a func-tion the area filling factor, i.e.ΦA = 4/3/ϑ.

where the efficiencyη = 0.1 andc is the speed of light. Thearea filling factor is inversely proportional to the transparencyof the torus (Fig. 5). Small area filling factors correspond totransparent tori.

We observe a tentative trend in the sense that in opaque torionly 0.1-10% of the torus mass accretion rate feeds the centralblack hole. On the other hand, if the torus is transparent,10-100% of the torus mass accretion rate is used to feed the centralengine.

The main differences between opaque (ϑ ∼ 1) and trans-parent tori (ϑ > 3) are the cloud collision rate and the cloudmean free path. Since there is no correlation with the collisionratetcoll = ϑ/Ω, we suspect the mean free path to be respon-sible for the accretion efficiency of the tori. The differentmeanfree paths between the clouds means that in transparent torithe AGN emission can reach all clouds of the torus, whereasin the opaque case it only reaches clouds at small galactocen-tric radii, i.e. close to the central source. Due to this radiationonly the densest clouds can survive, the less denser clouds be-ing evaporated by a photodissiciation or X-ray dissociation re-gion (PDR/XDR). Vollmer & Duschl (2001) calculated the lo-cation of the ionization fronts in the cloud of the circumnucleardisk in the Galactic Center. These clouds have about the samemass as the AGN torus clouds (Mcl ∼ 10 M⊙) and sizes of∼ 0.1 pc. Whereas these clouds are close to the shear limit,the AGN torus clouds are∼ 100 times denser. Note however,that in the Galactic Center the source of ionization is a centralcluster of about 40 O stars which is much weaker than an AGN.

Vollmer & Duschl (2001) showed that the cloud radius dueto the ionization front is given by

rcl = 3.645 1015J−

1

3

UV c−

4

3

i c8

3

s , (50)

whereJUV is the number of UV photons per cm2 and sec,ciis the sound speed in the ionized gas, andcs is the sound speedin the cloud. If one assumes that the cloud temperature is de-

termined by the external radiation fieldcs ∝ T1

2 ∝ J1

8

UV, the


cloud radius does not change with the cloud’s distance from thegalactic center.

We scale up the Galactic Center by a factor of104 andobtain ci = 7.6 km s−1, cs = 3 km s−1(R/(1 pc))−

1

4 , andJUV = 1.8 1016 cm−2s−1(R/(1 pc))−2 for a transparent torus.This leads to a cloud radius ofrcl = 0.03 pc, which is veryclose to the cloud radius obtained for our sample of AGN tori(Table 4). Thus, only dense, selfgravitating clouds can survivein an illuminated environment, i.e. a transparent torus. Ontheother hand, if the torus is opaque, less dense and larger cloudscan survive. Since we derive only the properties of the dens-est and most massive clouds in our model which determine thephysics of the outer torus, less dense and less massive cloudsare still consistent with the model as long as they do not domi-nate the disk mass. As a consequence, the transparency valuesderived for the individual AGNs and summarized in Table 4 areupper limits. We therefore suggest that towards the inner edgeof opaque tori clouds become less dense with densities closetothe shear limit. These clouds, however, once they arrive at theinner edge of the torus, are (i) destroyed easily by the influenceof shear and possible winds and are (ii) much easier ionized andevaporated by the AGN emission. In addition, for AGN close tothe Eddington limit Honig & Beckert (2007) showed that dustyclouds experience a strong radiation pressure. For transparenttori only dense and compact clouds smaller than the shear limitwill be found, while for opaque tori shadowing by clouds atthe inner edge allows larger clouds to survive at intermediatedistances.

This tentative picture leads to two predictions for AGN torithat might be verified in the future:

– the inner parts of opaque tori should harbor larger cloudsand should therefore have larger volume filling factors,

– the ratio of the torus mass accretion rate to the mass re-moved in winds generated at the inner boundary of torishould be lower for opaque tori.

9. Uncertainties

Up to this point we ignored any possible observational errors.In this section the influence of the quantities derived from ob-servations on the correlations shown in Fig. 2 and Fig. 5 is in-vestigated. We assume the following uncertainties:

– For the gas mass (Mgas) - about a factor 2, because of theuncertainty in mass distribution and hence in interpretingthe kinematics;

– peak star formation rate (Mpeak∗ ) - at least a factor 2-3 since

the extinction is not well constrained;– starburst age (tSB) - up to a factor 2, assuming that all diag-

nostics are explained by a single burst, which we reject forNGC3783;

– rotation velocity (vrot) - ∼ 40%, since it is the equivalentKeplerian rotation velocity, and depends on both the mea-sured dispersion and inclination-corrected velocity.

– turbulent velocity dispersion (vturb) - ∼ 20% assumingthat the velocity dispersion is not strongly affected by awind.

All uncertainties of our derived quantities are thus dominatedby the observational uncertainties onMgas and M∗. For thetorus mass accretion rate we have

M =128

π4fv2turbv

−2rotM

2gasM

−2dync

−4s ΩMcl . (51)

During the torus evolution the mass accretion rate is propor-tional to the square of the gas massM ∝ M2

gas. The gas massof the massive (Q ∼ 1) turbulent disk only depends on the peakstar formation rate

Mgas ∝ (Mpeak∗ )

5

11 . (52)

We took this gas mass as the initial gas mass of the collisionaltorus at the beginning of Phase II. Thus the ratio between themass accretion rate of the present torusM and that of the ini-tial collisional torusMold (Fig. 2) depends on the square ofthe present gas massMgas and on the inverse of the peak starformation rateMpeak

∗ :

M

Mold

∝M2

gas

Mpeak∗

. (53)

On the other hand, the ratio between the mass accretion rateonto the black holeMBH and the torus mass accretion rateMdepend on the AGN luminosityL and on the inverse of thesquare of the gas massMgas:

MBH

M∝

L

M2gas

. (54)

An underestimation of the gas mass, which we assumed to be10% of the dynamical mass, thus leads to a strong increase ofthe two ratios, while an underestimation of the peak star for-mation rate and the AGN luminosity leads to a decrease of thetwo ratios. In extreme cases both ratios can have errors up toafactor of10.

The starburst ages have uncertainties of a factor of 2. Thearea filling factor is proportional to the inverse of the gas massΦV ∝ M−1

gas (Eq. 35). Thus, the associated uncertainty is also afactor of 2.

Despite these relatively large uncertainties we believe thatthe correlations shown in Fig. 2 and 5 are real. All system-atic errors, like an overestimate of the gas mass, would alterbut not destroy the correlations, as long as the errors for dif-ferent targets are not random. However, to make our findingsmore robust more spectroscopic VLT SINFONI observationswith high spatial resolution of AGNs are needed. Most impor-tantly, future ALMA high resolution CO line observations arenecessary to determine the total gas mass of the tori with anuncertainty of∼ 10− 30%.

10. A holistic view of torus evolution in AGN

VLT SINFONI observations of the close environments (∼30 pc) of a sample of nearby AGNs by Davies et al. (2007)showed that thick gas tori and recent central starbursts withages smaller than100 Myr are ubiquitous. We compare differ-ent clumpy accretion disk models to these observations:


– fully gravitational turbulent disks where the turbulence ismaintained by the energy input from the gravitational po-tential via mass accretion,

– supernova and stellar wind driven turbulent disks (SNmodel) where the turbulence is maintained by stellar windsand supernova explosions,

– collisional disks where the orbital motion is randomized bypartially elastic collisions which also allow mass transportto the center and angular momentum redistribution.

Whereas the measured rotation velocity, turbulent velocity dis-persion, and gas mass tell us about the current state of the gastorus, the peak star formation rate and the age of the starburstprovide information on the past appearance of the torus. Weassume that the physical properties of the torus are mainly de-termined by external mass accretion from scales of∼ 100 pc.

The result of this work is a time sequence for the torus evo-lution. Present tori appear to be collisional and geometricallythick whereas the tori giving rise to the starburst in the pastare of turbulent nature and relatively thin. The torus evolutioncan be divided into 3 phases depending on the external massaccretion rate:

– Phase I: initial massive infall: formation of a massive tur-bulent stellar wind-driven gas disk withQ ∼ 1,

– Phase II: decreasing, but still high mass accretion rate: col-lisional thick torus,

– Phase III: decreasing, now low mass accretion rate: colli-sional thin torus.

Phase I is short (∼ 10 Myr). Once the SN explode, they removethe intercloud medium and clear the disk leaving behind onlythe densest clouds. The result is a collisional torus. All tori dis-cussed in this paper are interpreted to be in phase II. Thereforethis phase can last for more than100 Myr. The transparencyof these tori depend on their Toomre parameter, the turbulentvelocity, and the internal pressure of the clouds (temperatureand magnetic fields). If there is a second short massive infall,the torus can again switch into phase I. We suggest that thishas happened in the case of NGC 3783. On the other hand,an initial massive gas infall is not mandatory as the case ofCircinus illustrates. Once the external mass accretion rate hassignificantly decreased, the torus becomes thin and transparent.The Circumnuclear Disk in the Galactic Center represents thisphase.

In addition we conclude that

1. The massive turbulent stellar wind-driven gas disk (phaseI) gives rise to a starburst with a star formation rate thatequals the disk mass accretion rate.

2. The collisional torus also forms stars, but with an efficiencywhich is about10% that of the turbulent disk.

3. During the evolution of the thick collisional torus (phase II)the decrease of the mass accretion rate is slow (a factor of4 within 150 Myr).

4. The present collisional tori (phase II) do not significantlyobscure the stellar populations born during the recent star-burst. This is the case even for Sy2 nuclei.

5. We find a tentative correlation between the area filling fac-tor of the clouds and the ratio between the mass accretion

rate onto the central black hole and the torus mass accre-tion rate. AGNs with a high area filling factor lose morethan90% of the mass transported through the torus in athermal torus wind at the inner edge. We suggest that this isdue to shadowing of the gas clouds from the central AGNthroughout most the extended torus which allows the exis-tence of low density clouds whose densities are close to theshear limit.

Future VLT SINFONI and ALMA observations will benecessary to confirm our proposed torus evolution sequence.

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arxiv:0809.3397v1 [astro-ph] 19 sep 2008surprisingly,all molecular gas tori have high velocity...

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